Rocky Mountain Journal of Mathematics

Metric heights on an Abelian group

Charles L. Samuels

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Suppose $m(\alpha)$ denotes the Mahler measure of the non-zero algebraic number $\alpha$. For each positive real number $t$, the author studied a version $m_t(\alpha)$ of the Mahler measure that has the triangle inequality. The construction of $m_t$ is generic and may be applied to a broader class of functions defined on any Abelian group $G$. We prove analogs of known results with an abstract function on $G$ in place of the Mahler measure. In the process, we resolve an earlier open problem stated by the author regarding $m_t(\alpha)$.

Article information

Rocky Mountain J. Math., Volume 44, Number 6 (2014), 2075-2091.

First available in Project Euclid: 2 February 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R04: Algebraic numbers; rings of algebraic integers 11R09: Polynomials (irreducibility, etc.) 20K99: None of the above, but in this section
Secondary: 26A06: One-variable calculus 30D20: Entire functions, general theory


Samuels, Charles L. Metric heights on an Abelian group. Rocky Mountain J. Math. 44 (2014), no. 6, 2075--2091. doi:10.1216/RMJ-2014-44-6-2075.

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